Definition df-plpr 3958

Description: Define pre-addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126.

Assertion

Ref	Expression
df-plpr	|- +pR = {<.<.x, y>., z>. | ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +P. u), (v +P. f)>.))}

Distinct variable group(s):   x,y,z,w,v,u,f

Detailed syntax breakdown of Definition df-plpr

Step	Hyp	Ref	Expression
1		cplpr 3784	. 2 class +pR
2		vx	. . . . . . 7 set x
3	2	cv 1089	. . . . . 6 class x
4		cnp 3779	. . . . . . 7 class P.
5	4, 4	cxp 2408	. . . . . 6 class (P. X. P.)
6	3, 5	wcel 1092	. . . . 5 wff x e. (P. X. P.)
7		vy	. . . . . . 7 set y
8	7	cv 1089	. . . . . 6 class y
9	8, 5	wcel 1092	. . . . 5 wff y e. (P. X. P.)
10	6, 9	wa 196	. . . 4 wff (x e. (P. X. P.) /\ y e. (P. X. P.))
11		vw	. . . . . . . . . . . . 13 set w
12	11	cv 1089	. . . . . . . . . . . 12 class w
13		vv	. . . . . . . . . . . . 13 set v
14	13	cv 1089	. . . . . . . . . . . 12 class v
15	12, 14	cop 1810	. . . . . . . . . . 11 class <.w, v>.
16	3, 15	wceq 1091	. . . . . . . . . 10 wff x = <.w, v>.
17		vu	. . . . . . . . . . . . 13 set u
18	17	cv 1089	. . . . . . . . . . . 12 class u
19		vf	. . . . . . . . . . . . 13 set f
20	19	cv 1089	. . . . . . . . . . . 12 class f
21	18, 20	cop 1810	. . . . . . . . . . 11 class <.u, f>.
22	8, 21	wceq 1091	. . . . . . . . . 10 wff y = <.u, f>.
23	16, 22	wa 196	. . . . . . . . 9 wff (x = <.w, v>. /\ y = <.u, f>.)
24		vz	. . . . . . . . . . 11 set z
25	24	cv 1089	. . . . . . . . . 10 class z
26		cpp 3781	. . . . . . . . . . . 12 class +P.
27	12, 18, 26	co 3001	. . . . . . . . . . 11 class (w +P. u)
28	14, 20, 26	co 3001	. . . . . . . . . . 11 class (v +P. f)
29	27, 28	cop 1810	. . . . . . . . . 10 class <.(w +P. u), (v +P. f)>.
30	25, 29	wceq 1091	. . . . . . . . 9 wff z = <.(w +P. u), (v +P. f)>.
31	23, 30	wa 196	. . . . . . . 8 wff ((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +P. u), (v +P. f)>.)
32	31, 19	wex 678	. . . . . . 7 wff E.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +P. u), (v +P. f)>.)
33	32, 17	wex 678	. . . . . 6 wff E.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +P. u), (v +P. f)>.)
34	33, 13	wex 678	. . . . 5 wff E.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +P. u), (v +P. f)>.)
35	34, 11	wex 678	. . . 4 wff E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +P. u), (v +P. f)>.)
36	10, 35	wa 196	. . 3 wff ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +P. u), (v +P. f)>.))
37	36, 2, 7, 24	copab2 3002	. 2 class {<.<.x, y>., z>. | ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +P. u), (v +P. f)>.))}
38	1, 37	wceq 1091	1 wff +pR = {<.<.x, y>., z>. | ((x e. (P. X. P.) /\ y e. (P. X. P.)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = <.(w +P. u), (v +P. f)>.))}

This definition is referenced by:  addsrpr 3978

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